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Subcompletions of representable relation algebras

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The variety of representable relation algebras is closed under canonical extensions but not closed under completions. What variety of relation algebras is generated by completions of representable relation algebras? Does it contain all relation algebras? It contains all representable finite relation algebras, and this paper shows that it contains many non-representable finite relation algebras as well. For example, every Monk algebra with six or more special elements (called “colors”) is a subalgebra of the completion of an atomic symmetric integral representable relation algebra whose finitely-generated subalgebras are finite.

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Acknowledgements

The most direct inspiration for this work is [15]; see also [12, 13, 16, 19], and especially [14]. The method used in a proof of representability in [15] was embodied in the notion of flexible trio. A modification of a construction from [15] causes the finitely-generated subalgebras to be finite. Suggestions for the form and content of the Introduction came from the referee and the editors, to whom I express my thanks.

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  1. Department of Mathematics, Iowa State University, Ames, IA, 50011, USA

    Roger D. Maddux

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  1. Roger D. Maddux
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Correspondence to Roger D. Maddux.

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In memoriam, Bjarni Jónsson.

This article is part of the topical collection “In memory of Bjarni Jónsson” edited by J.B. Nation.

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Maddux, R.D. Subcompletions of representable relation algebras. Algebra Univers. 79, 20 (2018). https://doi.org/10.1007/s00012-018-0493-0

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